Failings and Fixes of EnKF for Nonlinear Dynamics

1. Failings and Fixes of EnKF for Nonlinear Dynamics 2009 EnKF Workshop Presenter: Al Reynolds Co-authors: Youdou Wang, Reza Tavakoli and Gaoming Li

2. Outline • Consistency issue • Consequences of limited ensemble size

3. Estimation with EnKS State Vector: m: Porosities, log-permeabilities, initial depths of fluid contacts. p: Reservoir simulation primary variables. d: Production data, BHP, GOR, WCT.

4. Introduction to EnKS Method Overcoming inconsistency requires iteration • Adjoint gradient based – Li & Reynolds (2007) • HIEnKS – Periodically rerun from time zero using latest ensemble of models, 2008 EnKF workshop, SPE 119056, 2009 RSS • EnRML, Gu and Oliver (2007).

5. Structure Map of PUNQ-S3 Fault, gas cap, strong aquifer. grid. Data: BHP GOR WCT Match to 4032 days

6. True GOC 20ft Prior Mean of GOC True OWC 20ft Prior Mean of OWC Examples Prior means of both fluid contact depths too deep Standard deviation: 20 ft • Simultaneously estimate horizontal and vertical log-permeabilities, porosities and depths of initial fluid contact.

7. Estimations of Fluid Contacts – Example A EnKS-HIEnKS EnKS

8. WCT Prediction – Example A EnKS-EnRML EnKS-HIEnKS EnKS Assimilation Rerun

9. Horizontal Permeabilities – Example A Truth EnKS EnKS-HIEnKS

10. Estimation of Fluid Contacts – Example A • On average EnKS gives the correct estimate, but variance of estimate is large. • EnRML and HIEnKS give similar results but HIEnKS requires less than one-half the computational cost.

11. Two sets of two-phase relative permeability curves (Seiler et al.,2009) oil-water : oil-gas :

12. Estimates from EnKS (Gray: prior; Blue: posterior; Green: mean of posterior; Red: true.)

13. Estimates from HIEnKS (Gray: prior; Blue: posterior; Green: mean of posterior; Red: true.)

14. Pro-11 WCT prediction obtained by rerunning from time zero (b) HIEnKS (a) EnKS Very scary! EnKS appears to be more consistent but gives a 11% over estimation of oil initially in place.

15. Consequences of Limited Ensemble Size • Limited degrees of freedom to assimilate data. Updated model is a linear combination of models in initial ensemble • Loss of rank (Lorenc - assimilation of perfect data leads to loss of rank, decrease in degrees of freedom) can lead to filter divergence. • Poor estimates of covariances due to sampling error. • Underestimation of variance; • Spurious correlations leading to error updates of components of state vector far from observation locations.

16. Mitigation of Sampling Errors in EnKF • Increase ensemble size. Expensive. How big? • Covariance inflation - ad hoc. • Covariance localization. As correlation length becomes large, it converges to normal EnKF. Schur product of a positive definite correlation matrix with a positive semi-definite matrix is positive definite. • Effectively zeros correlation between data and components of state vector far from an observational location; • Effectively lets individual components (or groups of components) be updated with a different linear combination of initial ensembles.

17. Covariance Localization (Schur product is element by element multiplication.)

18. Description of Brugge Benchmark Study • 139X48X9 gridblocks; • 10 injectors and 20 producers; • 3 ICVs per well.

19. Description of Brugge test case • Porosity, NTG, Sw, Kx, Ky and Kz are given with uncertainty; • PVT, RelPerms, Perforations, Free Water Levels, are provided without noise; • For each well, oil rate, water rate and FBHPs (every 30 days) are recorded; • Changes in saturation and pressure were Interpreted from seismic data changes in saturation and pressure over a 10 year period are given.

20. Data matches of oil rates for years 10 to 20 obtained by rerunning from time zero with ensemble of models obtained by assimilating data from 0-20 years

21. Comparison of RMSE

22. Prior mean Main correlation length=10,000 ft, Ensemble mean

23. The ensemble mean model of ln(kx)after assimilating data from years 0-20; Left: without localization; Right: localization.

24. Main correlation length: 10,000 ft 3,000 ft

25. Comments • Without localization, data matches obtained for years 10-20 are poor and result in essentially identical data predictions during years 10-20; obtained biased predictions and an underestimation of the uncertainty in predictions. • By using covariance localization in Brugge case, we are able to improve the data matches for the second ten years and also avoid making substantial changes in the geology far from the wells. • Moreover, realized NPV increased from $4.29 billion to $4.52 billion.